Schematic representation of a compartmental contagion process on a network.

<p>(A) Illustration of a contagion process evolving on a time-varying network. Nodes’ colors correspond to their current state; edges denote current contacts between nodes; edge colors correspond to: black: no contagion may take place over the edge, red: contagion takes place during the present time-step, and red-to-blue gradient: contagion is possible but does not take place. (B) Example: reaction types in the SIR model. (C) Spontaneous reaction: a node <i>i</i> may spontaneously transition from its current state <i>x</i><sub><i>i</i></sub> to <math><msubsup><mi>x</mi><mi>i</mi><mo>′</mo></msubsup></math> with rate λ<sub><i>m</i></sub>. (D) Contact-dependent reaction: a node <i>i</i> may transition from its current state <i>x</i><sub><i>i</i></sub> to <math><msubsup><mi>x</mi><mi>i</mi><mo>′</mo></msubsup></math> with rate λ<sub><i>m</i></sub> upon contact with the node <i>j</i> in state <i>x</i><sub><i>j</i></sub>. (E) Mixed transition: a node <i>i</i> may spontaneously transition from its current state <i>x</i><sub><i>i</i></sub> to another state, <math><msubsup><mi>x</mi><mi>i</mi><mo>′</mo></msubsup></math> with rate λ<sub><i>m</i></sub>; contact with another node <i>j</i>, in state <i>x</i><sub><i>j</i></sub>, may alter the transition rate of <i>m</i>, <math><mrow><msub><mo>λ</mo><mi>m</mi></msub><mo>→</mo><msubsup><mo>λ</mo><mrow><mi>m</mi></mrow><mo>′</mo></msubsup></mrow></math>. After the contact (<i>i</i>, <i>j</i>)<sub><i>t</i></sub> ends, the transition rate may revert to λ<sub><i>m</i></sub>, remain unchanged, or change to a third value.</p>